Dimension and entropy for a class of stochastic processes

Rudemo, Mats (1964) Dimension and entropy for a class of stochastic processes. A MAGYAR TUDOMÁNYOS AKADÉMIA MATEMATIKAI KUTATÓ INTÉZETÉNEK KÖZLEMÉNYEI, 9 (1-2). pp. 73-87.

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Abstract

In the following we shall give a definition of dimension and entropy for a class of stochastic processes with the property that the sample functions are step functions with probability one. Dimension and entropy together give a measure of the uncertainty associated with a random variable, or in our case a stochastic process, see [1] and [9]. In § 1 we give some examples of stochastic processes, whose sample functions a. s.² are step functions — such a process is called a purely discontinuous process, a PDP. Some known properties of the dimension and the entropy for random variables and vectors, needed in the following, are stated in § 2. In § 3 we define dimension and entropy for a class of PDP : s regarded on a finite interval (0, T). As an example the dimension and entropy of a Poisson process are calculated. The asymptotic T-dependence of the dimension is studied in § 4 for ergodic Markov chains with a finite state space and for renewal processes. For vector processes the corresponding definitions are made in § 5. An example with Poisson processes is discussed. Finally in § 6 we give a method of approximating by PDP : s stochastic processes whose sample functions are a. s. continuous. Especially the Brownian motion is discussed and the dimension of the approximating PDP is studied.

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